"Some of our Democratic friends are threatening to shut down the government this month without an agreement on [Deferred Action for Childhood Arrivals] DACA. That's simply a hysterical and cynical ploy, putting their party and agenda ahead of the nation," Cornyn said from the Senate floor.

He said negotiations on passing legislation to address the DACA program are "currently at an impasse. [Democrats] are jeopardizing the future fate of those DACA recipients."

The Trump administration announced earlier this year that it would end the DACA program, which allows immigrants brought into the country illegally as children to work and go to school without fear of deportation.

Democrats want a deal linking border security to the DREAM Act, that would give immigrants brought into the country illegally as children a path to citizenship.

"When we get senators who are willing to accept the principle of the DREAM Act ... If the senator from Texas will start with the DREAM Act we can then engage in additional conversations, including border security," Durbin said on Monday.

Durbin previously told The Washington Post that he is willing to vote to shut down the government without a deal on DACA.

But Republicans, as well as President Trump, said after a closed-door White House meeting last month that they would not accept a DACA deal as part of the government funding bill.

Democrats, however, believe they have leverage in government funding negotiations because Republicans will need their votes in the Senate, and potentially the House, to avoid a shutdown.